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September 25, 2007

Progic II

Posted by David Corfield

My research time at present largely consists of the time it takes me to travel to Canterbury and back. Here’s what I came up with on the subject of progic on yesterday’s trip down.

A probabilistic predicate, PP, I said back here, is a map from XX to [0,1][0, 1]. But another way to look at it is as a conditional probability distribution on the set 22 given XX. This is a map in the Kleisli category of the probability monad. In fact, our probabilistic predicate is a map P:X→2P: X \to 2. Now we can look at the composition of a distribution over XX and this PP.

Let’s take XX a set of dogs, Q(x)Q(x) is a probability distibution over XX, perhaps recording my degree of belief in which dog I saw just now as it dashed past. Then if P(x)P(x) is the predicate recording my belief as to whether xx is a poodle, then ∑xP(x)×Q(x)\sum_x P(x) \times Q(x) is my degree of belief that I just glimpsed a poodle.

This is one of Mumford’s random constants, i.e., probabilistic predicates over {*}, or maps in the Kleisli category 1→21 \to 2.

Now how much of all that stuff about functions between XX and YY and adjunctions that I mentioned before can we bring with us? First, why not have probabilistic functions, or in other words conditional distributions Own(y|x)Own(y | x), e.g., recording my uncertainty as to the owner of dog xx? Then for every probabilistic predicate on humans, I can pull it back to give a probabilistic predicate on dogs. E.g., if F(y)F(y) records my degree of belief that person yy is French, I have a degree of belief, ∑yF(y)⋅Own(y|x)\sum_y F(y) \cdot Own(y | x) that xx is owned by a Frenchman, and ∑x,yF(y)⋅Own(y|x)⋅Q(x)\sum_{x, y} F(y) \cdot Own(y | x) \cdot Q(x) that the dog I just glimpsed is owned by a Frenchman.

But can we form those adjunctions? An obvious choice of a structure to place on the probabilistic predicates on XX is a partial order where P≤QP \leq Q if and only if P(x)≤Q(x)P(x) \leq Q(x), for all x∈Xx \in X. So if ‘Poodle’ ≤\leq ‘owned by a Frenchman’, do we have something like ‘owner of at least one poodle’ ≤\leq ‘French’? It seems to me unlikely that there are mappings from probabilistic predicates over YY to those over XX, adjoint to this ‘multiplication’ by Own(y|x)Own(y | x).

Posted at September 25, 2007 9:34 AM UTC

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Re: Progic II

You might want to look into using the category of probability kernels (where the objects are measurable spaces, and the morphisms are probability kernels between these spaces, which can be thought of either as families of probability measures in the target space indexed by the domain, or as a linear map from probability measures in the domain to that in the target). It seems to be a fairly convenient category for managing concepts such as relative information, relative products, conditional expectation, independence, etc.; they also show up in learning theory. (Strangely, however, they seem to fail to show up much on the internet; I was unable to find a good introduction to them online, but see for instance page 20 of this book.)

[Hmm, weird, I can’t get that URL to show up at all. It was Kallenberg’s “Foundations of modern probability” on Google Books.]

Re: Progic II

Terry wrote:

You might want to look into using the category of probability kernels (where the objects are measurable spaces, and the morphisms are probability kernels between these spaces, which can be thought of either as families of probability measures in the target space indexed by the domain, or as a linear map from probability measures in the domain to that in the target).

This sounds exactly like the ‘Kleisli category’ or ‘category of free algebras’ that David and I are talking about!

Indeed there are always two equivalent ways to describe morphisms in such situations. The simpler way is the first one you mention. This is what I called a probabilistic morphism from XX to YY: a map from XX to the space of probability measures on YY:

X→fPYX \stackrel{f}{\to} P Y

But the second way makes it a bit more obvious how to compose these morphisms.

Re: Progic II

It’s nice to see progress on the progic project. Do you think it’s philosophically pukka enough to slip through the foundationalist filter into the vats of funding? Certainly both probability and logic still count as ‘philosophical’ — while the territories of geometry and group theory have somehow been lost to philosophers.

What follows is a brief attempt to understand, explain, and popularize what you’ve done so far.

First: when first learning monads, I found the terms ‘Kleisli category’ and ‘Eilenberg–Moore category’ tremendously unevocative and off-putting. Instead of talking about the ‘Eilenberg–Moore category’ of a monad, I wish people would talk about the ‘category of algebras’ of a monad — since that’s what it is. And, instead of talking about the ‘Kleisli category’ of the monad, I wish they’d talk about its ‘category of free algebras’. Let’s call a spade a spade!

Anyway:

You seem to be blending logic as done in the topos SetSet with ‘my’ probability monad
P:Set→SetP : Set \to Set
Or maybe you’re blending logic as done in the topos (?) of Polish spaces with ‘your’ probability monad
P:Pol→PolP : Pol \to Pol
I guess when making initial forays into the subject it’s not worth getting too specific about which formalism you’re using. The approach using sets has the advantage of being cheap and low-tech — everyone thinks they know what a set is. But eventually you’ll want to use the high-budget Polish space approach, to get Lebesgue measure on [0,1][0,1] into the game!

One advantage of keeping both options on the table is that it provides an excuse to imagine you’re working very generally, starting from any topos equipped with a ‘probability monad’. I’m not sure what a probability monad actually is at this level of generality. But that doesn’t matter! I’ll just take the formal structure of ‘topos with monad’ and use it to examine what you’ve done so far…

Given objects XX and YY in our topos, a probabilistic morphism from XX to YY is a morphism from XX to probability measures on YY:
f:X→PY f: X \to P Y

You’ve given two examples:

A point in XX is a morphism
f:1→X f: 1 \to X
so a probabilistic point is a morphism
f:1→PX f: 1 \to P X
that is, a probability measure on
XX.

A predicate on XX is a morphism
f:X→Ω f: X \to \Omega
where Ω\Omega is the subobject classifier. So, a probabilistic predicate of XX is a morphism
f:X→PΩ f: X \to P\Omega
In your examples Ω=2\Omega = 2 and PΩ=[0,1]P\Omega = [0,1],
so a probabilistic predicate on XX just assigns a probablity to each point of XX.

As you note, we can compose a probabilistic point and a probabilistic predicate and get a point in PΩP\Omega, that is, a probability. This is just the probability that our probabilistic predicate is true for our probabilistic point.

How can we compose probabilistic morphisms? We can do it thanks to the wonders of the Kleisli category… err, the category of free algebras of PP.

So: go ahead and blend everything you know about logic — formulated using topoi — with probability theory — formulated using the probability monad. A bunch of stuff will work just because you’ve got a topos with a monad on it. But at some point, you’re gonna want to know how the monad gets along with the topos structures: products, coproducts, exponentials, etc. What can you say about these:

P(X×Y)P(X \times Y)

P(X+Y)P(X + Y)

P(YX)P(Y^X)

For example, does the probability monad admit a strength? Is it monoidal? I think so.

Someone who actually knows topos theory, like Todd Trimble, could probably list three dozen ways that a monad can get along with a topos.

Re: Progic II

By the way, I know that in the comment above I sound like one of those pompous jerks who try to embed any new idea in some elaborate formalism just to make it harder to understand.

It’s just that people have put a lot of thought into topoi and monads, so it’s fun if you know about them to see how much of David’s progic project can be phrased in those terms — and doing this raises some new questions, namely, how the operations of logic (built into the topos) and probability theory (built into the monad) interact.

In short, smashing big abstract formalisms against each other can shoot off interesting sparks.

Re: Progic II

I’ve never tried to understand what ‘admitting a strength’ really means. Is there an easy example to get what it’s about?

This ‘getting along’ you mention between monad and topos is the tricky part. In the case of logic over Sets you have Pred(X+Y)(X + Y) = Pred (X)×(X) \times Pred (Y)(Y), where ‘Pred’ gives you the predicates on a set.

And, as I mentioned, substitution from predicates of one set to another by way a function, preserves both join and meet.

But in the case of the probability monad, distributions over (X+Y)(X + Y) don’t neatly match up with distributions over XX and over YY. I suppose you could say they give you a distribution over XX and one over YY and a parameter to tell you the mixture. But even there, if XX or YY is outside the support, then there’s trouble.

Re: Progic II

I’m not too clear on why people invented strong monads in the first place. But, it seems pretty likely that the probability monad admits a strength, since there’s an obvious map

X×PY→P(X×Y) X \times PY \to P(X \times Y)

sending a point x∈Xx \in X and a probability measure μ\mu on YY to the probability measure δx⊗μ\delta_x \otimes \mu on X×YX \times Y.

I haven’t directly checked that it satisfies all the axioms, but I bet it does… especially since the
Wikipedia article defines commutative strong monads on symmetric monoidal categories and says “One interesting fact about commutative strong monads is that they are ‘the same as’ symmetric monoidal monads.”

While I again haven’t bothered to check it, it seems pretty obvious that the probability monad is monoidal, meaning we have a very well-behaved map

PX×PY→P(X×Y) P X \times P Y \to P(X \times Y)

sending a probability measure μ\mu on XX and a probability measure ν\nu on YY to the probability measure μ⊗ν\mu \otimes \nu on X×YX \times Y.

In short, I get the impression that while there are lots of subtly different conditions saying that a monad gets along with tensor products, the probability monad satisfies all of these with respect to the cartesian product.

Re: Progic II

The Kleisli category of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad. The canonical adjunction between C and the Kleisli category is a monoidal adjunction with respect to this monoidal structure.

Re: Progic II

Robin wrote:

A symmetric monoidal closed category CC is enriched over itself, and a strong monad is just a monad on CC in the 2-category of CC-enriched categories. It turns out that a monad admits a tensorial strength just when it “is” strong in this sense.

Well heck — then why didn’t they just come out and admit it?

However, I’m not sure I understand. You seem be giving a slick definition of a strong monad on a symmetric closed monoidal category. But they seem to be defining a strong monad on an arbitrary monoidal category.

I don’t see how having a strong monad on a monoidal category is enough to force that category to be symmetric and closed.
So, maybe your slick definition of strong monad only applies to symmetric closed monoidal categories, not more general one? But maybe you’re hinting that the extra generality is not worth worrying about?

With all this talk of monoidal monads, and strengths, and the like, we keep bumping into a very elegant idea: taking any doctrine and looking at monads in that doctrine! I’d like to better understand the full power of this idea. For example: I guess in nice cases we get a category of algebras for the monad which again lies in the doctrine?

(Of course, a doctrine can itself be seen as a categorified monad. This is just an example of how, past a certain level of abstraction, every concept interacts with every other in mind-bendingly circular ways.)

Re: Progic II

However, I’m not sure I understand. You seem be giving a slick definition of a strong monad on a symmetric closed monoidal category. But they seem to be defining a strong monad on an arbitrary monoidal category.

I think you understand perfectly, John, though I should have explained myself more carefully. If you think about C-enriched monads on a symmetric monoidal closed C, then (provided you’re clever enough, like Anders Kock) you eventually realise that they can be characterised in a way that only uses the tensor product, via the “tensorial strength”. Then you can turn around and say, let’s use this characterisation as a definition! So you have then generalised the notion to an arbitrary monoidal category.

Of course this is a common trick in mathematics: take some definition, find a characterisation that makes sense in a more general setting, then turn around and use that characterisation as a generalised definition that makes sense in the wider setting.

I’d dearly love to think about your second question; but I have to submit my thesis on Monday, so I’d better get back to it. :-)

Re: Progic II

PS. There’s also an ‘abstract nonsense’ way of generalising the notion of self-enriched monad to arbitrary monoidal categories. I don’t know that this gives the same result as the tensorial strength definition, but I’d be surprised if it didn’t. (Perhaps Todd Trimble knows?)

We know fron Brian Day’s thesis work that every monoidal category (every promonoidal category, even) embeds into a monoidal biclosed one: a (pro)monoidal structure on CC induces a monoidal biclosed structure on [C,Set]op[C, Set]^{op}. So that means that a (pro)monoidal category CC is [C,Set]op[C, Set]^{op}-enriched in a canonical way.

What does it mean for a monad on CC to be enriched in [C,Set]op[C, Set]^{op}? I bet it’s equivalent to having a tensorial strength.

Re: Progic II

No, I’d never thought about that, but since you gave me a shout-out, and have other more important things to do yourself (best wishes for your defense by the way!), sure, let me have a go at it.

I’ll first expand just a little on what Robin said before, about the connection between enrichment and strength. If VV is a closed monoidal category and we start with a strong functor T:V→VT: V \to V, then we get a VV-enrichment of TT:

where we have used in turn the unit of the adjunction (−)⊗Tv⊣(−)Tv(-) \otimes T v \dashv (-)^{T v}, the strength on TT, and the evaluation or counit of (−)⊗v⊣(−)v(-) \otimes v \dashv (-)^v. In the other direction, starting with a VV-enriched T:V→VT: V \to V, we get a strength:

where I’ll leave it to the reader to figure out what’s going on at each step.
So, as Robin said, any functor T:Set→SetT: Set \to Set, being Set-enriched, carries a canonical strength. (Although my saying ‘canonical’ is overkill – there’s just one strength, period. I think.)

(here the ⊗C\otimes_C is really ‘module composition’) and now we are considering what it would mean if T!T_! were SetCopSet^{C^{op}}-enriched, or equivalently admits a SetCopSet^{C^{op}}-strength,

F⊗T!(G)→T!(F⊗G),F \otimes T_!(G) \to T_!(F \otimes G),

where the tensor is Day convolution. Since T!T_! is cocontinuous and ⊗\otimes is separately cocontinuous in each of its arguments, and every FF is a colimit of representables, such a strength is (by naturality) uniquely determined by how it behaves on representables F=hom(−,c)F = hom(-, c), G=hom(−,d)G = hom(-, d). Here

Re: Progic II

Oh, I forgot, Robin said “monad”, not just a functor. It all works out fine, just the same, i.e., Robin spoke the truth: a strong monad TT on a monoidal category CC is tantamount to the left Kan extension T!:SetCop→SetCopT_!: Set^{C^{op}} \to Set^{C^{op}} being a SetCopSet^{C^{op}}-enriched monad.

Re: Progic II

All monads on Set are strong: they can’t help it. This is obvious if (and only if?) you know the original motivation for the definition of strength (by Anders Kock).

A symmetric monoidal closed category C is enriched over itself, and a strong monad is just a monad on C in the 2-category of C-enriched categories. It turns out that a monad admits a tensorial strength just when it “is” strong in this sense.

But “Set-enriched” is vacuous: all monads (functors, natural transformations) are Set-enriched! So every Set-monad has a canonical strength.

Re: Progic II

Combining probability theory and topos theory is not pompous. It is actually a very old idea invented by Dana Scott (for Boolean valued models) in his proof of the independence of the Continuous Hypothesis.
Scott’s idea is simple: if we add random real numbers, then the continuum does not have cardinality aleph_1. The proof, of course, was a bit more difficult … :-)

Matthew Jackson has recently picked up the topic again:
http://www.lawrence.edu/fast/jacksoma/research.html

For instance, showing that both measures and measurable functions are internal real numbers. This is similar to the fact that the interpretation of the reals in the topos of sheaves over a topological space are precisely the continuous real-valued functions.